Honors Math 2
Rational and Irrational Numbers
(Section 1.05)
Name:
Date:
Warm Up
!
Remember that a rational number is a number that you can express as ! , where a and b
are integers and 𝑏 ≠ 0.
1. Show that each number is rational by writing it as a fraction with an integer in the
numerator and the denominator.
a. 1.341
b. 1.3 + 2.8
!
c. 5 !
2. Prove that 2 is irrational using a proof by contradiction. A few steps are done for
you.
Assume 2 is rational.
If 2 is rational, then 2 =
(Now square both sides of your equation and see if you can reach a contradiction)
What happens when you add, subtract, or multiply integers? Is your result always an
integer?
Now let’s consider rational numbers. What happens when we add, subtract,
multiply, or divide rational numbers?
Prove that the sum of two rational numbers is rational.
You’ll always get a rational number when you add, subtract, multiply, or divide 2 rational
numbers (as long as you are not dividing by zero). You’ll prove some of these facts on
the homework.
Now let’s consider irrational numbers. What happens when we add, subtract,
multiply, or divide two irrational numbers.
Find a pair of irrational numbers whose difference is irrational.
Find a pair of irrational numbers whose difference is rational.
Find a pair of irrational numbers whose sum is irrational.
Find a pair of irrational numbers whose sum is rational.
Find a pair of irrational numbers whose product is irrational.
Find a pair of irrational numbers whose product is rational.
Find a pair of irrational numbers whose quotient is irrational.
Find a pair of irrational numbers whose quotient is rational.
Theorem: The sum of a rational number and an irrational number is irrational.
Let 𝑥 = 𝑞 + 𝑟 where q is rational and r is irrational.
Assume x is rational.
Theorem: The product of a nonzero rational number and an irrational number is
irrational.
Let 𝑥 = 𝑞𝑟 where 𝑞 is rational and 𝑟 is irrational.
Homework:
Prove that 5 is irrational.
Textbook problems: p. 26 #6-8, 12-16